Hermann Weyl’s reflections on mathematics, science, knowledge, and consciousness are among the most important and interesting of the 20th century. These two books very valuably re-present him to us, and Mind and Nature comes with a very helpful sympathetic introduction by Peter Pesic. Perhaps for that reason it may be useful to offer an alternative point of view here.

Very crudely, there are two perspectives on the nature of knowledge, especially scientific knowledge. The realist holds that in the end our knowledge of the world is determined by what is out there in the world; the idealist holds that really all we deal with is the ideas in our minds. Realists would like to get themselves, and all observers, out of the way and give neutral descriptions; idealists find subjectivity inescapable. Realists have tended to favour dry, scientific language, even boring academic prose; idealists will embrace the lofty, poetic, and mysterious. Realists cluster in Anglo-Saxon environments; idealists in Continental Europe and in Paris-focussed centres elsewhere. Realists find idealists somewhat unreadable; idealists read the likes of Fichte, Hegel, and Heidegger. Most scientists when they turn philosophical may wish to tell us that general relativity and quantum mechanics paint a picture of the world utterly unlike Newtonian mechanics, but they are realists at heart; Weyl began as an idealist with a strong grounding in the work of Fichte, although, Pesic tells us, he never got on with the ideas of Heidegger, and drifted towards the realists in his American years.

Weyl was well aware that his Continental background could make him sound strange to American years; he sounds stranger today. The Yale lectures “The Open World’’ (1932) have as their theme: “How does the divine manifest itself in nature?’’ He distinguished two traditional answers: ‘’the ether is the omnipresence of God’’ and “the mathematical lawfulness of nature is the revelation of divine reason”. Weyl favoured the latter interpretation, provided it was understood in the light of contemporary physics.

In the long essay “Mind and Nature” (1934) he first presented what he took to be the realist point of view, then took up an epistemological position that questioned realism in principle while extending “the theory of the subjectivity of the sense qualities [to] the perceptual forms of space and time”. He argued that scientific cognition does not describe states of affairs but constructs symbolic representations of the world of appearances. The critique of realism is based on accounts of perception, especially colour perception. The epistemological response Weyl gave begins: “The world does not exist independently but only for a consciousness. We realise by epistemological refection that a quality like color can be given only in consciousness, in sensation; it has no meaning at all to assign color to a thing in itself as a property detached from consciousness. This insight by which naïve realism is in principle overcome constitutes the beginning of all our philosophizing. It opposes the epistemological standpoint of idealism to that of realism in teaching us to consider the immediate data of consciousness as primary, the world of objects as secondary.”

If you think this has it exactly backwards this section of the essay is an interesting one that discusses how our concepts are built into a perceptions, but it still ends up by claiming that “What compels us to refer our immediate experience to an objective symbolic world is, originally, no doubt, our belief in the validity of recollection’’, a belief that is rooted in “that knowing, acting, caring, existence that is utterly different from the existence of things”. Where is scientific method in all of this? Weyl answered that the mathematical scientist presumes a free realm of possibilities, from which he or she selects those free of arbitrary variation between observers. Some of these are then handled symbolically for the purpose of preservation and communication, and these symbols are embedded in an ordered manifold of possibilities.

Perhaps recognising that this formulation is not altogether clear, Weyl asked us to think about a comet. On the one hand it makes its way from today’s position to tomorrow’s by really performing its motion. We, on the other hand, find its position tomorrow by means of a complicated symbolic calculation. “What is it”, he asked, “that this symbolic process of the astronomer has in common has in common with the real process of the comet?” Without offering a simple answer, Weyl claimed that his arguments had shown that scientific knowledge is conditioned by the circumstance that the world, the goal of all our scientific endeavors, is not one existing in itself, but arises from and exists only by means of the meeting of subject and object.”

It is of course true that the idealist can have fun with the realist as the foundations of physics change. The poor realist is left stranded by most major advances in physics: special and general relativity and quantum mechanics falsify most of what they previously said. But the idealist should not laugh; he or she is caught out too. And if they say too loudly that they are only changing ideas, which are in their minds and theirs to change, they risk sounding as if nothing they say has much claim on anyone else’s attention — it is just their ideas and they could be deluded. Weyl’s position was better than this, of course, but his much-praised tendency to poetic language does not always help him here. Idealism can appear as one of those subversive philosophies young adults and precocious teenagers embrace because it offers a thrill: You used to think things were like that but in fact they are like this. Realism in all its forms is beset with problems, but Weyl can sound as if the thrill of rejecting naïve realism has got him no further than elaborating an equally dubious, and sometimes frankly murky, idealism. A better way forward would be to start where Weyl in the 1930s and 1940s stopped, and make some attempt to keep subject and object together.

One place where Weyl’s idealism most disconcerts the conventional mathematician is his philosophy of the infinite. Whatever the ins and outs of his Brouwerian phase in the 1920s, which he only ended when he found he could not think like an intuitionist and think mathematically about quantum mechanics, he remained doubtful about transfinite set theory because he disputed the mind’s ability to survey a completed infinite set. This went with his unorthodox attitude to the continuum, which he refused to accept was properly considered in what he called its ‘atomized’ form as a set of points. Now that even the official line in set theory admits many possible different universes of sets and different axiomatisations of both set theory and logic, it is worth remarking that Weyl did not advocate an unorthodox view alongside the emerging orthodoxy; he thought the orthodox position was wrong, meaningless, and a dangerous act of self-deception. This opinion probably cannot be refuted; it has its exponents today, but its highly marginal status suggests that on this occasion idealism has not read the human mind correctly.

Weyl wrote when ideas, especially those in quantum mechanics, were in flux. It was natural to feel that even the best scientific theories offered only possibilities that had been articulated, perhaps, in a mathematically sound way. But scientists seldom stay long with the world of mere representation. They cease to say “It is as if there is a particle with behaviour expressed by these equations” and they start to say simply that there is such-and-such a particle. Now that generations have grown up with what is, after all, our best attested theory of physics, it seems reasonable to rephrase our epistemology of quantum mechanics to admit a degree of realism, to tone down the subjective element, and to think more rigorously about what it is to understand something in the world outside ourselves. Weyl was open to this, as tentative remarks in his rich volume Philosophy of Mathematics and Natural Science indicate; these books have much to stimulate the philosopher of science today, perhaps because of, rather than in spite of the heavily idealistic leanings. More importantly, they still have much to offer the philosophically minded physicist and mathematician.

Jeremy Gray is a Professor of the History of Mathematics at the Open University, and an Honorary Professor at the University of Warwick, where he lectures on the history of mathematics. In 2009 he was awarded the Albert Leon Whiteman Memorial Prize by the American Mathematical Society for his work in the history of mathematics. His latest book is Plato’s Ghost: The Modernist Transformation of Mathematics.